Three-dimensional slow Rossby waves in rotating spherical density-stratified convection

We develop a theory of three-dimensional slow Rossby waves in rotating spherical density stratified convection. The Rossby waves, with frequencies which are much smaller than the rotating frequency, are excited by a nonaxisymmetric instability from the equilibrium based on the developed convection. These waves interact with the inertial waves and the density stratified convection. The density stratification is taken into account using the anelastic approximation for very low-Mach-number flows. We study long-term planetary Rossby waves with periods which are larger than two years. We suggest that these waves are related to the southern oscillation and El Niño. The El Niño is an irregularly periodical variation in winds and sea surface temperatures over the tropical Pacific ocean, while the southern oscillation is an oscillation in surface air pressure between the tropical eastern and the western Pacific ocean. The strength of the southern oscillation is characterized by the southern oscillation index (SOI). The developed theory is applied for the interpretation of the observed periods of the SOI. This study demonstrates a good agreement between the theoretical predictions and the observations.

Figure 1

Time dependence of the southern oscillation index (SOI) after 5 month window averaging, where the time is measured in years, $t=0$ corresponds to the year 1878, and the total time interval of the observations of the SOI is 138 years. The data are taken from [29].

Figure 2

The spectrum $E_{\mathrm{SOI}}\left(f\right)$ of the SOI. The frequency is measured in the units of inverse years. The shown periods of oscillations (13.88; 6.61; 4.96; 3.96; 3.08; 2.57) are measured in years.

Figure 3

The period $T_R=2\pi /\left|{\omega }_R^{\left(3D\right)}\right|$ (measured in years) of the slow 3D Rossby waves vs the characteristic scale $L_z/H_{\rho }$, where $L_z=\pi /k$ for $m=1$ (solid) and $m=2$ (dashed). The stars (for $m=1$) and squares (for $m=2$) correspond to the observed periods of the SOI; see Fig. 2. The vertical sizes of these observed modes, $L_z=7.3;10.6;12.4;16.3$ km, determine the wave numbers $k$ of the slow 3D Rossby waves; see Eqs. (23) and (25).

Figure 4

The characteristic time $T_{\mathrm{inst}}={\gamma }_R^{-1}$ (measured in years) of the excitation of the 3D Rossby waves vs the characteristic scale $L_z/H_{\rho }$ of the convective cells for $m=1$ (solid) and $m=2$ (dashed).